Pythagorean Expectation for Bettors: Points to Wins, and the Exponent for Each Sport
A win total is a claim about points. A team priced at 10.5 wins in the NFL carries an implied scoring margin. The board never shows the margin, but you pay for it. This article shows the formula, the exponent each sport uses, the point value of one win, and a table of the margin behind every common win total.
The main finding: the exponent debate is a distraction in baseball and football. The exponent moves a 17-game team by under half a win. Scoring margin, schedule, and one-score luck move the same team by two wins or more. Basketball is the exception, and the NBA table below shows why.
The Formula
Bill James built the formula for baseball in the early 1980s. Expected win percentage equals points scored raised to a power, divided by the sum of points scored and points allowed, each raised to the same power. James started with an exponent of 2.
Pythagorean expected win percentage
W% = PF^x / (PF^x + PA^x)
Expected wins = W% × games
Example: 391 points for, 347 against, 17 games, x = 2.37
391^2.37 / (391^2.37 + 347^2.37) = 0.5703
0.5703 × 17 = 9.69 expected wins
A team with 391 points for and 347 against has a 44-point margin. The formula says the margin is worth 9.7 wins, not the 12 the same team might have posted.
The Exponent by Sport
Each sport needs its own exponent because points arrive at different rates and with different randomness. Fewer scoring events per game means more chance in each result and a lower exponent.
| Sport | Exponent | Source |
|---|---|---|
| MLB | 2.00 (original), 1.83 (refined) | The MLB glossary reports Baseball-Reference uses 1.83 |
| NFL | 2.37 | Schatz (2003), cited in Dayaratna and Miller |
| NBA | 14 | Basketball-Reference, per Nylon Calculus and Bill James Online |
| NBA (alternate) | 16.5 | ESPN, per the same Nylon Calculus piece |
Sources disagree on who found the NFL value. The 2025 Frontiers in Sports paper credits Daryl Morey. Dayaratna and Miller credit Aaron Schatz. Both land on 2.37.
The NBA range is wide. A Journal of Sports Analytics study of 1993 to 2014 seasons found the best fit slightly above 14, and describes the accepted range as 14 to 17.
Why baseball sits near 1.8
Steven Miller derived the formula in "A Derivation of the Pythagorean Won-Loss Formula in Baseball" (Chance, 2007). He modeled runs scored and runs allowed as independent Weibull variables with a shared shape parameter. His fits to 2004 American League data gave a shape near 1.79, close to the 1.83 refinement. Dayaratna and Miller then argued the same model fits hockey about as well as baseball.
Does the Exponent Matter? A Stress Test
You pick an exponent, then you pick a team. Which choice moves the answer more? The tables hold the team fixed and change only the exponent.
| Sport and team | Exponent | Expected wins |
|---|---|---|
| NFL, 391 PF / 347 PA, 17 games | 2.00 | 9.51 |
| NFL, same team | 2.37 | 9.69 |
| NFL, same team | 2.70 | 9.86 |
| MLB, 800 RS / 700 RA, 162 games | 1.80 | 90.7 |
| MLB, same team | 1.83 | 90.8 |
| MLB, same team | 2.00 | 91.8 |
| NBA, 118 PF / 111 PA, 82 games | 14.0 | 57.6 |
| NBA, same team | 16.5 | 60.1 |
The NFL exponent swing from 2.0 to 2.7 costs 0.35 wins. The MLB swing from 1.8 to 2.0 costs 1.1 wins out of 162. Neither changes a bet. The NBA choice between 14 and 16.5 moves the same roster by 2.5 wins. A season win total prices in half-win steps, so 2.5 wins is the whole market. In the NBA, state your exponent before you quote a number.
Points per Win
The formula has a clean slope near .500. Write the ratio as r = PF / PA. Then W% = r^x / (1 + r^x). At r = 1 the derivative is x / 4. A team with a small margin D in a league where each side scores S per game gains about x × D / (4 × S) in win percentage per game.
Points of margin per win (derived from the slope)
Win % per point of per-game margin = x / (4 × S)
NFL: 2.37 / (4 × 22) = 0.0269, so one win (0.0588 of 17 games) needs 2.18 points per game, or 37 points a season
NBA: 14 / (4 × 114) = 0.0307, so one point per game is worth 2.5 wins, or 32.6 points a season per win
MLB: 1.83 / (4 × 4.5) = 0.1017, so one win needs 9.8 runs a season
These are your inputs: S is points per team per game. Change S and the answer moves. A 22-point NFL average and a 4.5-run MLB average are assumptions you should replace with the current league figure. The values above show the method, and the calculator below does the arithmetic for your numbers.
Win Totals Translated to Margin
Reverse the formula. Take a win total, convert to win percentage, and solve for the PF to PA ratio: r = (W% / (1 - W%))^(1/x). Fix PF + PA at league average, and the margin falls out.
The tables ignore the juice on a posted total. A line priced -115 on both sides carries a vig, so the true median sits a little off the posted number. Treat each row as a rough read.
NFL (x = 2.37, 22 points per team per game, 17 games)
| Win total | Win % | Implied PF | Implied PA | Margin | Per game |
|---|---|---|---|---|---|
| 6.5 | 38.2% | 336 | 412 | -75 | -4.4 |
| 7.5 | 44.1% | 355 | 393 | -37 | -2.2 |
| 8.5 | 50.0% | 374 | 374 | 0 | 0.0 |
| 9.5 | 55.9% | 393 | 355 | +37 | +2.2 |
| 10.5 | 61.8% | 412 | 336 | +75 | +4.4 |
| 11.5 | 67.6% | 432 | 316 | +115 | +6.8 |
Each win adds 37 to 40 points of margin. The 8.5 row is the break-even: with 17 games a team cannot win exactly 8.5, so the market splits the difference.
MLB (x = 1.83, 9.0 combined runs per game, 162 games)
| Win total | Win % | Margin per game | Season run margin |
|---|---|---|---|
| 65.5 | 40.4% | -0.95 | -154 |
| 72.5 | 44.8% | -0.52 | -84 |
| 81.5 | 50.3% | +0.03 | +5 |
| 90.5 | 55.9% | +0.58 | +94 |
| 98.5 | 60.8% | +1.07 | +174 |
NBA (x = 14, 228 combined points per game, 82 games)
| Win total | Win % | Margin per game |
|---|---|---|
| 30.5 | 37.2% | -4.27 |
| 38.5 | 47.0% | -0.99 |
| 46.5 | 56.7% | +2.20 |
| 54.5 | 66.5% | +5.57 |
| 62.5 | 76.2% | +9.48 |
Use It: Market Margin Versus Last Year's Margin
Take a hypothetical NFL team. Last year the team went 12-5 with 391 points for and 347 against. The formula says 9.69 wins, so the record ran 2.3 wins ahead of the margin. The exponent range from the stress test moves the gap between 2.1 and 2.5 wins. The verdict holds under every exponent.
Now suppose the team opens at 10.5 wins. The table row for 10.5 shows a +75 margin. Last year's margin was +44. The market assumes the team adds 31 points of margin over the full season, or 1.8 points per game. You now have a specific question to answer. Did the roster change by 1.8 points per game? Quarterback, offensive line, and defensive turnover answer the question. The record does not.
This article makes no regression claim. Secondhand sources say teams who beat their Pythagorean wins by a full game fall back the next year. Other sources dispute the size of the effect. None of the sources checked here publishes the data to size the effect. Compare the market margin to the roster, not to a fixed discount.
Fit Exponent Versus Forecast Exponent
The Journal of Sports Analytics study separates two jobs. Describing a season already played favors an exponent near 14 in the NBA. Predicting the next stretch favors a lower one. The authors report the forecasting exponent falls substantially below 14.
The reason is regression. Margins in a short window carry noise. A lower exponent pulls every team toward .500 and works as a built-in shrink. The paper's lower value is not quoted in the sources checked here, so the table below uses 10 as an illustration only.
| NBA team, 118 PF / 111 PA | Exponent | Win % | Wins in 82 |
|---|---|---|---|
| Describe the past | 14 | 70.2% | 57.6 |
| Illustrative forecast | 10 | 64.9% | 53.2 |
Pick the tool by the question. A recap uses the fit exponent. A win-total bet uses a lower one, and the right value depends on how many games sit in your sample.
Calculators
Points to expected wins
Win total to implied margin
Limits
- The formula reads points, not schedule. Add a strength of schedule adjustment before you compare two teams.
- Garbage-time scoring in football and basketball inflates margin for blowout winners.
- Injuries change the future team. Last year's margin describes the roster you no longer have.
- Ties and overtime rules shift hockey and football totals. This article gives no hockey exponent because the sources checked here do not agree on one.
- The table margins assume league-average scoring. A slow team and a fast team with the same win total carry different point margins.
Rules to Keep
- Use 1.83 for MLB, 2.37 for NFL, and 14 for NBA when you describe a finished season.
- In the NFL and MLB, ignore exponent debates. The swing is under half a win in football and about one win in baseball.
- In the NBA, name your exponent. The 14 versus 16.5 gap is 2.5 wins.
- One NFL win equals about 37 points of season margin. One MLB win equals about 10 runs. One NBA point of per-game margin equals about 2.5 wins.
- Convert every win total to an implied margin, then ask what changed on the roster to earn the difference from last year.
Sources
- Miller, S. J. (2007). A Derivation of the Pythagorean Won-Loss Formula in Baseball. Chance 20(1).
- Dayaratna, K. D. and Miller, S. J. The Pythagorean Won-Loss Formula and Hockey.
- MLB Glossary, Pythagorean Winning Percentage.
- Journal of Sports Analytics, NBA exponent study of 1993 to 2014 seasons.
- Nylon Calculus, Over and under-performing expected win totals.
- Bill James Online, article 1146, on the NBA exponent.
- Frontiers in Sports and Active Living, 2025, article 1638446.